bibkey: kaygorodov2026evolution
authors: Ivan Kaygorodov; Cándido Martín González; Antonio Viruel
year: 2026
title: "Evolution algebras: a survey of results and a list of open problems"
doi: 10.48550/arXiv.2610.22248
url: https://arxiv.org/abs/2610.22248v1
claim: "Conjecture 4 asks for the simple complex evolution-algebra automorphism maximum max{n!, n(2^n-1)}; Theorem 4 and Conjecture 3 concern perfect algebra automorphisms; Question 46 asks whether dense square forces uniqueness of a Hilbert natural orthonormal basis; Question 14 asks whether an infinite natural-basis quotient can coexist with finite full automorphisms; Question 31 asks about realizing arbitrary linear algebraic groups as full evolution-algebra automorphism groups or group schemes."
strata_touched:
  - D5/S3/QuadraticForms/EvolutionAlgebras/SimpleEvolutionAutomorphismMaximum
  - D5/S3/QuadraticForms/EvolutionAlgebras/FiniteAutomorphismsInfiniteNaturalBases
  - D5/S3/QuadraticForms/EvolutionAlgebras/HilbertNaturalBasisNonuniqueness
  - D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionAutomorphismControl
  - D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionBasisTransport
  - D5/S3/QuadraticForms/EvolutionAlgebras/NilpotencySpectrumRefutation
  - D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionAutomorphismTangentBridge
  - D5/S3/QuadraticForms/EvolutionAlgebras/AlgebraicGroupRealizationRefutation
license: citation-only
triage: anchor

Evolution algebra automorphisms and ordinary nilpotency

Multiplication is bilinear and may be nonassociative. Theorem 4 supplies monomiality for perfect evolution algebras. Section 2.1.6 describes the diagonal arrow equations. Section 2.3, Conjecture 3 asks for the maximum automorphism order over all perfect complex evolution algebras of each positive dimension, including attainment, six specified small values and the factorial tail. Monomiality and finite diagonal groups are credited supporting results, not a solution of this conjecture. The quotient-control bound does not assert the sharp maximum, attainment or an exclusive extremizer classification.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2610.22248
• URL: https://arxiv.org/abs/2610.22248v1
• Text: https://arxiv.org/html/2610.22248v1
• Locations: Theorem 4; section 2.1.6 and Remark 19; section 2.3, Conjecture 3; Question 33 for the point-group sequence onto the permutation image.

The group-scheme sequence in equation (2) has no terminal surjectivity claim onto the full graph automorphism group. The formal model uses the actual image and the actual block-action kernel.

Hilbert natural-basis uniqueness

Section 2.2.8, Question 46, page 26 asks: “If the square of a Hilbert evolution algebra is dense, is its natural orthonormal basis unique up to permutation and unimodular scalars?” Section 1.8 and the cited Hilbert evolution algebra framework allow real or complex scalars, complete orthonormal natural bases, and continuous left multiplication. Here the square is the algebraic span of all pair products and density is norm density. The subsequent sentence about bounded derivations, automorphisms and spectral theory is a separate request. A complex counterexample refutes the first universal assertion for the full real-or-complex class. The first interrogative is a question in this source, not a theorem attributed to its authors.

Natural bases with finite full automorphisms

Section 2.2.2, Question 14 asks: “Describe the set of natural bases (up to permutation and scaling) of a non-degenerate evolution algebra. When is it finite? Can it be infinite for an algebra whose automorphism group is finite?” The selected assertion is the entire final sentence. Section 1.1 permits an arbitrary field, assumes finite dimension unless otherwise stated, and identifies nondegeneracy with zero annihilator. Aut denotes all field-valued linear multiplication automorphisms. Complex scalars give a permitted existence witness. Basis equivalence allows arbitrary permutations and independent nonzero scalars, without quotienting by algebra automorphisms.

The two-dimensional algebra and its finite full group are prior ingredients from arXiv:1807.02362v1 and arXiv:2303.00862v1. The Q14 endpoint combines these with a separated infinite family in the exact basis quotient. It does not answer the first two interrogatives or assert global priority.

Ordinary nilpotency spectrum

Section 1.1 defines ordinary lower powers by A^1=A and A^m=sum_{i=1}^{m-1} A^i A^{m-i}. The products are bilinear and need not be associative or power-associative. Section 2.3, Conjecture 9 states that over each infinite field an n-dimensional nilpotent evolution algebra of index k+1 exists exactly when k=2^a for 0<=a<=n-1, or k has at least two nonzero binary digits and floor(log_2 k)+s_2(k)<=n-1. A natural basis requires zero mixed products; zero natural squares are allowed.

The fifteen-dimensional rational algebra in D5/S3/QuadraticForms/EvolutionAlgebras/NilpotencySpectrumRefutation has ordinary index 256. At k=255, the logarithm and binary digit count are 7 and 8, whose sum exceeds 15-1. The construction and its proof are project results, while the question and definitions are from this survey.

Simple-algebra automorphism maximum

Conjecture 4, section 2.3, page 28 states: “Over ℂ, the maximal order of the automorphism group of a simple n-dimensional evolution algebra is max{n!, n(2ⁿ−1)}.” The dimension is positive. The same paragraph identifies the cyclic product and generic complete-support products as attaining families.

The source’s Theorem 6 uses simplicity in the usual ideal sense. Its cited Cabrera Casado, Siles Molina and Velasco source, arXiv:1601.01832v2, section 4, page 18, states: “We recall that an algebra A is simple if A² ≠ 0 and 0 is the only proper ideal.” The formal predicate quantifies over every complex vector subspace absorbing multiplication on both sides. The full group consists of all complex linear multiplication automorphisms, without an associativity or multiplicative-unit assumption.

The uniform maximum proof and the explicit I+J attaining family are repository results. Strong connectivity makes the successor quotient one cycle. A supported permutation gives equal positive quotient fibers. The joint group bound reduces to h(2ʰ−1)(k!)ʰ with n=hk; the factorial comparison and its boundary cases give the asserted maximum. Classification of all extremizers and attainment over other fields remain open here.

Algebraic-group realization

Question 31 (Realisation of algebraic groups), section 2.2.5, page 24: “Let G be a linear algebraic group over 𝕜. Is there a finite-dimensional evolution 𝕜-algebra A with Aut(A) ≅ G(𝕜) as abstract groups? with Aut(A) ≅ G as algebraic groups? with 𝐀𝐮𝐭(A) ≅ G as group schemes?”

The second and third interrogatives are the selected assertions. Over Complex, they are encoded by polynomial-closed subgroups of Matrix.GeneralLinearGroup and natural equivalences with all multiplication-preserving linear automorphisms after extension to every commutative unital coefficient algebra. This includes coefficient algebras with nilpotents. Finite-dimensional algebras are represented on Fin d → Complex, with an existential natural basis and no nondegeneracy or perfectness assumption. The SL3 obstruction is a repository-derived consequence of the semisimple derivation classification and the tangent construction. The full SLn family and the abstract-group question are not formalized here. The literature screen recorded in issue #21840 is partial; exhaustive priority and journal/preprint equivalence remain ASSUMED-UNVERIFIED.